In the case of normally distributed \(k\) -dimensional phenotypes, we characterize the population's phenotype statistics by the average \({\bf x}=[ x_1,x_2,\dots , x_k ]^\top \)  and a symmetric covariance matrix \(K\) , where the diagonal elements \(K_{ii}({\bf x})\)  indicate the variance of the \(i^{th}\)  phenotype and the off-diagonal entries \(K_{ij}({\bf x})\)  indicate the covariance between different phenotypes.
To model the evolution of such multi-variate phenotype, we consider the three primary evolutionary forces: natural selection, mutations and genetic drift. The effect of selection on the mean phenotype is proportional to the covariance between fitness and phenotype within a population [1]}. For Gaussian distributed phenotypes, the change in mean phenotype \(d{\bf x}\)  over a short time interval \({\text{d}}t\)  simplifies to a stochastic process [2]},
\(d{\bf x}= (K \cdot \nabla F + \nabla M) {\text{d}}t +\frac{1}{N} \Sigma \cdot {\text{d}}{\bf W}\) 
